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Theorem pellexlem3 35746
Description: Lemma for pellex 35750. To each good rational approximation of  ( sqr `  D
), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
Assertion
Ref Expression
pellexlem3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Distinct variable group:    x, D, y, z

Proof of Theorem pellexlem3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 10637 . . . 4  |-  NN  e.  _V
21, 1xpex 6614 . . 3  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 4914 . . 3  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  C_  ( NN  X.  NN )
42, 3ssexi 4541 . 2  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V
5 simprl 772 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  a  e.  QQ )
6 simprrl 782 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  0  <  a )
7 qgt0numnn 14779 . . . . . . . 8  |-  ( ( a  e.  QQ  /\  0  <  a )  -> 
(numer `  a )  e.  NN )
85, 6, 7syl2anc 673 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (numer `  a )  e.  NN )
9 qdencl 14769 . . . . . . . 8  |-  ( a  e.  QQ  ->  (denom `  a )  e.  NN )
105, 9syl 17 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (denom `  a )  e.  NN )
118, 10jca 541 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) )
12 simpll 768 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  D  e.  NN )
13 simplr 770 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  -.  ( sqr `  D )  e.  QQ )
14 pellexlem1 35744 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  -.  ( sqr `  D )  e.  QQ )  ->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
1512, 8, 10, 13, 14syl31anc 1295 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
16 simprrr 783 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
17 qeqnumdivden 14774 . . . . . . . . . . . 12  |-  ( a  e.  QQ  ->  a  =  ( (numer `  a )  /  (denom `  a ) ) )
1817oveq1d 6323 . . . . . . . . . . 11  |-  ( a  e.  QQ  ->  (
a  -  ( sqr `  D ) )  =  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )
1918fveq2d 5883 . . . . . . . . . 10  |-  ( a  e.  QQ  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  =  ( abs `  ( ( (numer `  a )  /  (denom `  a )
)  -  ( sqr `  D ) ) ) )
2019breq1d 4405 . . . . . . . . 9  |-  ( a  e.  QQ  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
215, 20syl 17 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
2216, 21mpbid 215 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
23 pellexlem2 35745 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )  ->  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) )
2412, 8, 10, 22, 23syl31anc 1295 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) )
2511, 15, 24jca32 544 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
)  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
2625ex 441 . . . 4  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  QQ  /\  (
0  <  a  /\  ( abs `  ( a  -  ( sqr `  D
) ) )  < 
( (denom `  a
) ^ -u 2
) ) )  -> 
( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
27 breq2 4399 . . . . . 6  |-  ( x  =  a  ->  (
0  <  x  <->  0  <  a ) )
28 oveq1 6315 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  ( sqr `  D ) )  =  ( a  -  ( sqr `  D ) ) )
2928fveq2d 5883 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  ( sqr `  D ) ) )  =  ( abs `  ( a  -  ( sqr `  D
) ) ) )
30 fveq2 5879 . . . . . . . 8  |-  ( x  =  a  ->  (denom `  x )  =  (denom `  a ) )
3130oveq1d 6323 . . . . . . 7  |-  ( x  =  a  ->  (
(denom `  x ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
3229, 31breq12d 4408 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
)  <->  ( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
) ) )
3327, 32anbi12d 725 . . . . 5  |-  ( x  =  a  ->  (
( 0  <  x  /\  ( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
3433elrab 3184 . . . 4  |-  ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  <->  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
35 fvex 5889 . . . . 5  |-  (numer `  a )  e.  _V
36 fvex 5889 . . . . 5  |-  (denom `  a )  e.  _V
37 eleq1 2537 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( y  e.  NN  <->  (numer `  a )  e.  NN ) )
3837anbi1d 719 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
y  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  z  e.  NN ) ) )
39 oveq1 6315 . . . . . . . . 9  |-  ( y  =  (numer `  a
)  ->  ( y ^ 2 )  =  ( (numer `  a
) ^ 2 ) )
4039oveq1d 6323 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( (
y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )
4140neeq1d 2702 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  <->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0
) )
4240fveq2d 5883 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) ) )
4342breq1d 4405 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( ( abs `  ( ( y ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) )  < 
( 1  +  ( 2  x.  ( sqr `  D ) ) )  <-> 
( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
4441, 43anbi12d 725 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
4538, 44anbi12d 725 . . . . 5  |-  ( y  =  (numer `  a
)  ->  ( (
( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
46 eleq1 2537 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( z  e.  NN  <->  (denom `  a )  e.  NN ) )
4746anbi2d 718 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a )  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) ) )
48 oveq1 6315 . . . . . . . . . 10  |-  ( z  =  (denom `  a
)  ->  ( z ^ 2 )  =  ( (denom `  a
) ^ 2 ) )
4948oveq2d 6324 . . . . . . . . 9  |-  ( z  =  (denom `  a
)  ->  ( D  x.  ( z ^ 2 ) )  =  ( D  x.  ( (denom `  a ) ^ 2 ) ) )
5049oveq2d 6324 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =  ( ( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )
5150neeq1d 2702 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  <->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 ) )
5250fveq2d 5883 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) ) )
5352breq1d 4405 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) )  <->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
5451, 53anbi12d 725 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
5547, 54anbi12d 725 . . . . 5  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
)  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
5635, 36, 45, 55opelopab 4723 . . . 4  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  <->  ( (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
(denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
5726, 34, 563imtr4g 278 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ->  <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
58 ssrab2 3500 . . . . . 6  |-  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  C_  QQ
59 simprl 772 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6058, 59sseldi 3416 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  QQ )
61 simprr 774 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6258, 61sseldi 3416 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  QQ )
6335, 36opth 4676 . . . . . . 7  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b
) ,  (denom `  b ) >.  <->  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )
64 simprl 772 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (numer `  a
)  =  (numer `  b ) )
65 simprr 774 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (denom `  a
)  =  (denom `  b ) )
6664, 65oveq12d 6326 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  ( (numer `  a )  /  (denom `  a ) )  =  ( (numer `  b
)  /  (denom `  b ) ) )
67 simpll 768 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  e.  QQ )
6867, 17syl 17 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  ( (numer `  a )  /  (denom `  a )
) )
69 simplr 770 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  e.  QQ )
70 qeqnumdivden 14774 . . . . . . . . . 10  |-  ( b  e.  QQ  ->  b  =  ( (numer `  b )  /  (denom `  b ) ) )
7169, 70syl 17 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  =  ( (numer `  b )  /  (denom `  b )
) )
7266, 68, 713eqtr4d 2515 . . . . . . . 8  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  b )
7372ex 441 . . . . . . 7  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) )  -> 
a  =  b ) )
7463, 73syl5bi 225 . . . . . 6  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  ->  a  =  b ) )
75 fveq2 5879 . . . . . . 7  |-  ( a  =  b  ->  (numer `  a )  =  (numer `  b ) )
76 fveq2 5879 . . . . . . 7  |-  ( a  =  b  ->  (denom `  a )  =  (denom `  b ) )
7775, 76opeq12d 4166 . . . . . 6  |-  ( a  =  b  ->  <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >. )
7874, 77impbid1 208 . . . . 5  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
7960, 62, 78syl2anc 673 . . . 4  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
8079ex 441 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  /\  b  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) } )  ->  ( <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) ) )
8157, 80dom2d 7628 . 2  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V  ->  { x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
824, 81mpi 20 1  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031   <.cop 3965   class class class wbr 4395   {copab 4453    X. cxp 4837   ` cfv 5589  (class class class)co 6308    ~<_ cdom 7585   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   QQcq 11287   ^cexp 12310   sqrcsqrt 13373   abscabs 13374  numercnumer 14761  denomcdenom 14762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-numer 14763  df-denom 14764
This theorem is referenced by:  pellexlem4  35747
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